Continuous Phase Modulation (CPM) is a bandwidth, energy and battery efficient digital modulation technique, exhibiting smooth transitions in its phase, which conveys the information content. Smooth phase transitions enable bandwidth efficiency gains by virtue of concentrating the signal's power in a compact spectral band, allowing adjacent waveforms to be packed closer together in the frequency domain. The constant envelope property of CPM leads to significant savings in DC (direct current) power, which translates into battery efficiency for mobile digital communication terminals, such as handsets or satellites.
To achieve the phase smoothness, most CPM techniques employ phase smoothing pulses such as raised cosine (RC) or rectangular (REC) pulses, whose length (L) may be several integer multiples of symbol duration (T), and acts as the modulation memory. This boils down to intentionally introducing intersymbol interference, as multiple pulses occupy one symbol interval. Besides the phase smoothing pulse, the two other free parameters managing the characteristics of a CPM signal are the modulation index h, and the alphabet size, M. The modulation index h is described as the ratio of the frequency deviation to the frequency of the modulating wave, and defined to be hk/p such that k and p are two mutually prime integers. As h gets smaller, the distance among the CPM symbols is decreased, which results in an even more compact spectra at the expense of reduced resilience to error. M on the other hand determines the variety of information messages to be transmitted. As M gets higher, transmission signals are selected from a richer set, offering greater error resilience.
A CPM signal can be expressed as
                              z          ⁡                      (                          t              ,              α                        )                          =                  exp          ⁢                      {                          j              ⁢                                                          ⁢              2              ⁢                                                          ⁢              π              ⁢                                                          ⁢              h              ⁢                                                ∑                  n                                ⁢                                                      α                    n                                    ⁢                                      q                    ⁡                                          (                                              t                        -                        nT                                            )                                                                                            }                                              (        1        )            where t denotes time, and T denotes the symbol duration, both measured in seconds, αn∈{±1, ±3, . . . , ±(M−1)} is the M-ary message symbol, and α stands for the history of message symbols such that α=[αn, αn−1, . . . ]. q(.) is the phase smoothing response of a CPM waveform, and it is related to the underlying frequency pulse, f(.) with the following relation:q(t)=∫−∞tf(τ)dτ  (2)
Unfortunately the optimal CPM demodulator has a demodulation complexity that varies exponentially in the pulse length. Specifically, an optimal CPM demodulator requires 2pML front end matched filters [1, 2], and a detector with pML−1 states in its trellis. This high demodulation complexity renders the implementation of highly spectrally efficient CPM schemes unrealistic on the state-of-the-art digital hardware platforms.
The front end filtering in a digital communication receiver enables to maximize the signal-to-noise ratio (SNR) by matching the receiver filter to the transmission filter. The trellis structure in a digital communication system on the other hand describes the evolution of the transmission symbols in terms of nodes and branches of a graph. Each node in the trellis graph is termed a state, which are interconnected with branches. The combination of the previous trellis state and the present information symbol activate a branch in the trellis graph, which results in the emission of a new transmission symbol.
In a digital communication receiver, the number of states in the trellis is a primary indicator of complexity, as the detector in the receiver has to calculate the a posteriori probability of each state transition if the Maximum a posteriori Probability (MAP) criteria is invoked as the objective of the detection. Likewise, the detector has to calculate the likelihood of each state transition if the Maximum Likelihood (ML) detection criteria is chosen. Then the detector selects the most probable, or the most likely branch for the MAP, and the ML criteria, respectively, to declare a decision. Two iterative, and popular algorithms to realize the MAP and ML detection are the BCJR [3] and the Viterbi algorithms [4], respectively.
To reduce the computational complexity of a CPM demodulator, various CPM decomposition techniques have been proposed, among which the work of Laurent [5], and that of Mengali and Morelli [6] are the most prominent techniques from a complexity-reduction standpoint. Laurent's decomposition expresses a CPM waveform as a superposition of Pulse Amplitude Modulated (PAM) components for M=2, i.e., binary CPM. Laurent's seminal work for binary CPM is generalized to an arbitrary CPM alphabet size by Mengali and Morelli [6]. According to Laurent, Mengali and Morelli's method, a CPM waveform can be approximated as
                              z          ⁡                      (                          t              ,              α                        )                          ≈                              ∑                          k              =              0                                      F              -              1                                ⁢                                    ∑              n                        ⁢                                          a                                  k                  ,                  n                                            ⁢                                                g                  k                                ⁡                                  (                                      t                    -                    nT                                    )                                                                                        (        3        )                                          Q          ⁢                      =            Δ                    ⁢                      2                          L              -              1                                      ,                  P          ⁢                      =            Δ                    ⁢                      ⌈                                          log                2                            ⁢              M                        ⌉                          ,                  F          ⁢                      =            Δ                    ⁢                                    Q              P                        ⁡                          (                              M                -                1                            )                                                          (        4        )            where F is the number of PAM components in the decomposition, αk,n denote CPM pseudosymbols, and gk(.) denote the Laurent pulses. The sign  denotes the mathematical definition of a parameter, and the operation ┌.┐ denotes the ceiling operation, which rounds its argument to the nearest integer above its current value. Generation of CPM pseudosymbols, αk,n with respect to the actual message symbols, αn, and the calculation of the Laurent pulses, gk(t) are presented in detail in the papers [5] for binary CPM, and in [6] for M-ary CPM.
Laurent-Mengali-Morelli decomposition can be utilized both at a transmitter and a receiver, which leads to a matched reception. In this case, since this decomposition is an approximation, there may be slight fluctuations in the radio frequency (RF) envelope of the transmitted signal, yielding a quasi-constant envelope signal, rather than a strictly constant envelope signal. Alternatively, Laurent-Mengali-Morelli decomposition may be utilized only at a receiver, while transmitter emits true CPM signal with constant envelope, leading to a mismatched reception.
An important observation related to gk(.) is that the main pulse, i.e., g0(t), carries a significant portion of the signal energy in a binary CPM realization. This observation leads to various simplifications of CPM waveforms by simply truncating the PAM decomposition to a single pulse representation for binary CPM, where only the main pulse, g0(t) is retained in the decomposition, and the rest of the pulses are discarded [7, 8]. Similarly, for M-ary CPM, the first 2P−1 pulses convey most of the signal energy. Following from this, for M-ary CPM, a principal pulses approximation is developed in [9-13], retaining the first 2P−1 pulses in the Laurent-Mengali-Morelli decomposition and dismissing the rest of the pulses.
As the state-of-the-art low complexity CPM demodulator design, the work of Colavolpe et al. [9-13] utilizes the principal pulse approximation of CPM, which yields a near-optimal demodulation performance for L=2. In particular, for L=2, the method presented in [9-13] employs 2P−1 matched filters at the front end, and an ML or MAP detector with a p-state trellis. For L=3 however, the complexity of the method presented in [9-13] is increased such that an ML or MAP or detector with pM-state trellis is required. The performance of this method, however, deteriorates significantly for L>3, and the complexity reduction gains are compromised. Hence the state-of-the-art low complexity CPM demodulator [9-13] is adequate only for L=2.
Another reduced complexity CPM demodulator design [7] is developed only for the binary CPM, (i.e., M=2), which utilizes a minimum mean square error (MMSE) filter, followed by a Viterbi detector.
Yet another reduced complexity CPM demodulator is proposed in [14], which however concerns only non-coherent reception.
What is needed in the art is a method and apparatus to simplify the coherent CPM demodulator design for all CPM configurations and arbitrary values of h, M, and L to achieve high spectral efficiencies with a near-optimal, or reasonable performance with low complexity, so that efficient signal processing algorithms and hardware may be designed. This need is seen for both constant or quasi-constant envelope CPM schemes.